Full Idea
If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
Gist of Idea
If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology
Source
E.J. Lemmon (Beginning Logic [1965], 2.4)
Book Reference
Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.80
A Reaction
The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.